Wyrównanie sieci niwelacyjnej jednakowo dokłądnej
Repery nawiązania: 4, 5, 6
Repery niewiadone (wyznaczane): 1, 2, 3					u=3
Spostrzeżenia: L1, L2, L3, L4, L5, L6, L7, L8, L9					n=9
Spostrzeżenia nadliczbowe: nn=n-u					nn=6
Wysokości reperów nawiązania					Różnice wysokości
Lp.	H [m]				Lp.	od	do	DH [m]
4	230.387				L1	4	1	0.934
5	231.650				L2	1	5	0.354
6	235.075				L3	4	2	2.170
					L4	2	6	2.511
					L5	5	3	0.955
					L6	3	6	2.495
					L7	1	2	1.243
					L8	1	3	1.257
					L9	3	2	-0.017
Repery niewiadone (wyznaczane)
H1w = H10 + dh1
H2w = H20 + dh2
H3w = H30 + dh3
1. Obliczenie współrzędnych przybliżonych Hi0
H10 = H4 + L1
H20 = H4 + L3
H30 = H5 + L5
2. Utworzenie układu równań poprawek
H4 + L1 + v1 = H1
H4 + L1 + v1 = H10 + dh1
H4 + L1 + v1 = H4 + L1 + dh1
v1 = dh1
H1 + L2 + v2 = H5
H10 + dh1 + L2 + v2 = H5
H4 + L1 + dh1 + L2 + v2 = H5
v2 = -dh1 - H4 + H5 - L1 - L2
v2 = -dh1 + w2
H4 + L3 + v3 = H2
H4 + L3 + v3 = H20 + dh2
H4 + L3 + v3 = H4 + L3 + dh2
v3 = dh2
H2 + L4 + v4 = H6
H20 + dh2 + L4 + v4 = H6
H4 + L3 + dh2 + L4 + v4 = H6
v4 = -dh2 - H4 + H6 - L3 - L4
v4 = -dh2 + w4
H5 + L5 + v5 = H3
H5 + L5 + v5 = H30 + dh3
H5 + L5 + v5 = H5 + L5 + dh3
v5 = dh3
H3 + L6 + v6 = H6
H30 + dh3 + L6 + v6 = H6
H5 + L5 + dh3 + L6 + v6 = H6
v6 = -dh3 - H5 + H6 - L5 - L6
v6 = -dh3 + w6
H1 + L7 + v7 = H2
H10 + dh1 + L7 + v7 = H20 + dh2
H4 + L1 + dh1 + L7 + v7 = H4 + L3 + dh2
v7 = -dh1 + dh2 - L1 + L3 - L7
v7 = -dh1 + dh2 + w7
H1 + L8 + v8 = H3
H10 + dh1 + L8 + v8 + H30 + dh3
H4 + L1 + dh1 + L8 + v8 + H5 + L5 + dh3
v8 = -dh1 + dh3 - H4 + H5 - L1 + L5 - L8
v8 = -dh1 + dh3 + w8
H3 + L9 + v9 = H2
H30 + dh3 + L9 + v9 = H20 + dh2
H5 + L5 + dh3 + L9 + v9 = H4 + L3 + dh2
v9 = dh2 - dh3 + H4 - H5 + L3 - L5 - L9
v9 = dh2 - dh3 + w9
Układ równań poprawek
v1	=	dh1
v2	=	-dh1			+ w2
v3	=		dh2
v4	=		-dh2		+ w4
v5	=			dh3
v6	=			-dh3	+ w6
v7	=	-dh1	+ dh2		+ w7
v8	=	-dh1		+ dh3	+ w8
v9	=		dh2	- dh3	+ w9
Macierzowy układ równań
v = A * x + w
v	=	A			*	x	+	w
v1	=	1	0	0	*		+	0.000
v2		-1	0	0				-0.025
v3		0	1	0				0.000
v4		0	-1	0		dh1		0.007
v5		0	0	1		dh2		0.000
v6		0	0	-1		dh3		-0.025
v7		-1	1	0				-0.007
v8		-1	0	1				0.027
v9		0	1	-1				-0.031
3. Utworzenie układu równań normalnych
(AT * A) * x + (AT * w) = 0
AT									*	A			=	AT * A
									*	1	0	0	=
										-1	0	0
										0	1	0
1	-1	0	0	0	0	-1	-1	0		0	-1	0		4	-1	-1
0	0	1	-1	0	0	1	0	1		0	0	1		-1	4	-1
0	0	0	0	1	-1	0	1	-1		0	0	-1		-1	-1	4
										-1	1	0
										-1	0	1
										0	1	-1
AT									*	w	=	AT * w
									*	0.000	=
										-0.025
										0.000
1	-1	0	0	0	0	-1	-1	0		0.007		0.005
0	0	1	-1	0	0	1	0	1		0.000		-0.045
0	0	0	0	1	-1	0	1	-1		-0.025		0.083
										-0.007
										0.027
										-0.031
AT * A			*	x	+	AT * w	=	0
4	-1	-1	*	dh1	+	0.005	=	0
-1	4	-1		dh2		-0.045		0
-1	-1	4		dh3		0.083		0
4. Rozwiązanie układu równań normalnych
(AT * A)-1 * (AT * w) = -x
(AT * A)-1			*	AT * w	=	-x
0.300	0.100	0.100	*	0.005	=	0.005
0.100	0.300	0.100		-0.045		-0.005
0.100	0.100	0.300		0.083		0.021
dh1 =	-0.005
dh2 =	0.005
dh3 =	-0.021
5. Obliczenie wyrównanych wysokości reperów
H1 = H4 + L1 + dh1
H2 = H4 + L3 + dh2
H3 = H5 + L5 + dh3
H1 =	231.316
H2 =	232.562
H3 =	232.584
6. Obliczenie spostrzeżeń wyrownanych
Liw = Li + vi
v = A * x + w
A			*	x	=	A * x	+	w	=	v
1	0	0	*		=	-0.005	+	0.000	=	-0.005
-1	0	0				0.005		-0.025		-0.020
0	1	0				0.005		0.000		0.005
0	-1	0		-0.005		-0.005		0.007		0.002
0	0	1		0.005		-0.021		0.000		-0.021
0	0	-1		-0.021		0.021		-0.025		-0.004
-1	1	0				0.010		-0.007		0.003
-1	0	1				-0.016		0.027		0.011
0	1	-1				0.026		-0.031		-0.005
L	+	v	=	Lw
0.934	+	-0.005	=	0.929
0.354		-0.020		0.334
2.170		0.005		2.175
2.511		0.002		2.513
0.955		-0.021		0.934
2.495		-0.004		2.491
1.243		0.003		1.246
1.257		0.011		1.268
-0.017		-0.005		-0.022
7. Kontrola wyrównania
Liw = Liobl
Liobl = Hi+1 - Hi
Lp.	Hi+1	Hi	Liobl		Liw	Liobl
L1	231.316	230.387	0.929		0.929	0.929
L2	231.65	231.316	0.334		0.334	0.334
L3	232.562	230.387	2.175		2.175	2.175
L4	235.075	232.562	2.513		2.513	2.513
L5	232.584	231.65	0.934		0.934	0.934
L6	235.075	232.584	2.491		2.491	2.491
L7	232.562	231.316	1.246		1.246	1.246
L8	232.584	231.316	1.268		1.268	1.268
L9	232.562	232.584	-0.022		-0.022	-0.022
8. Zestawienia
8a. Zestawienie wysokości wyrównanych					8b. Zestawienie spostrzeżeń wyrównanych
Lp.	nr	H [m]			Lp.	ciąg	Liw
1	1	231.316			L1	4-1	0.929
2	2	232.562			L2	1-5	0.334
3	3	232.584			L3	4-2	2.175
4	4	230.387			L4	2-6	2.513
5	5	231.650			L5	5-3	0.934
6	6	235.075			L6	3-6	2.491
					L7	1-2	1.246
					L8	1-3	1.268
					L9	3-2	-0.022